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Probability theory is all about measuring how likely an event will occur. When dealing with probability of events, we focus on calculating the likelihood of specific events happening.
For example, when we roll a single die, each face, or outcome, has an equal chance of occurring. The probability of any particular face landing up is \(rac{1}{6}\).
If we define events based on certain conditions, like having numbers less than 4 (Event \( A \)) or numbers less than or equal to 2 (Event \( B \)), we can calculate the probability by counting the favorable outcomes and dividing by the total number of possible outcomes.
This is why the probability of \( B \) is \( \frac{2}{6} \).To summarize, the probability of an event \( E \) happening is given by \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in the sample space}}. \]

Conditional probability helps us understand the likelihood of an event occurring, given that another event has already occurred.
In simple terms, it's about narrowing down the possible outcomes based on additional information. It is denoted as \( P(A \mid B) \), which reads as "the probability of \( A \) given \( B \)."Using our die example, if we know that event \( B \) has occurred (observing a number less than or equal to 2), and we want to find the probability of event \( A \) (observing a number less than 4), we consider only those outcomes that are in \( B \).
Since \( B \) is a subset of \( A \), the occurrence of \( B \) ensures \( A \), making \( P(A \mid B) = 1 \).The formula for conditional probability is: \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \]provided \( P(B) eq 0 \). This formula shows how to compute the likelihood of event \( A \) when we know \( B \) has happened.

The sample space is a fundamental concept in probability theory. It represents the set of all possible outcomes of a random experiment.
In terms of tossing a die, the sample space \( S \) is \({1, 2, 3, 4, 5, 6}\), capturing every possible result when the die is rolled.The sample space is essential because it serves as the reference set for calculating probabilities of events.
For instance, with a die, each number has a probability of \( \frac{1}{6} \), derived from the sample space.
The completeness of the sample space allows for the precise computation of probabilities for any defined event.Remember, without defining the sample space, gauging probabilities becomes ambiguous as we lack the full picture of possible outcomes. This is why understanding the sample space is the first step in approaching any probability problem.

Events can interact with each other in ways that can be measured by intersection and union, two key concepts in probability theory.The intersection of two or more events (denoted by \( A \cap B \)) refers to outcomes that are common to all the events involved.
For example, the intersection of events \( A \) and \( B \) is the set \({1, 2}\), thus \( P(A \cap B) = \frac{1}{3} \) because these are shared outcomes.Union, on the other hand, combines the outcomes from either or both events. It is denoted by \( A \cup B \).
The union of events \( A \) and \( C \) (outcomes: \( 1, 2, 3, 4, 5, 6 \)) covers all possible outcomes, so \( P(A \cup C) = 1 \).These concepts help in understanding how events overlap or spread, which is crucial for broader probability assessments.